203 research outputs found
Novel soliton solutions to the Atangana-Baleanu fractional system of equations for the ISALWs
This work deals the construction of novel soliton solutions to the Atangana-Baleanu (AB) fractional system of equations for the ion sound and Langmuir waves by using Sardar-subequation method (SSM). The outcomes are in the form of bright, singular, dark and combo soliton solutions. These solutions have wide applications in the arena of optoelectronics and wave propagation. The bright solitons will be a vast advantage in controlling the soliton disorder, dark solitons are also beneficial for soliton communication when a background wave exists and singular solitons only elaborate the shape of solitons and show a total spectrum of soliton solutions created from the model. These results would be very helpful to study and understand the physical phenomena in nonlinear optics. The performance of the SSM shows that this is powerful, talented, suitable and direct technique to discover the exact solutions for a number of nonlinear fractional models
New solitary wave and computational solitons for Kundu-Eckhaus equation
The goal of this research is to find novel optical solutions to the Kundu-Eckhaus equation, which possess crucial roles in the field of nonlinear optics. A collective variable (CV) strategy is adopted to solve governing equation including the Raman effect and quintic nonlinearity. This method is a suitable to deal with both conservative and non-conservative systems by exposing a set of equations of motion regardless of nonlinearities or dissipative components. The parameters employed in this approach are chirp, temporal position, phase, amplitude, frequency and width, namely, collective variables. The fourth order Runge-Kutta technique is a well-known numerical scheme that aims towards the solution of the resulting system of ordinary differential equations representing the variables involved in the pulse ansatz. This technique presents the evolution of pulse parameters with regard to propagation variables. The graphical profiles at suitable values of pulse parameters are also provided. The unified technique is also applied to find soliton solutions. The obtained solution is a periodic solitary wave, showed graphically. The results developed in this article are found to be new in the literature and the approach utilized, can be applied to solve a variety of nonlinear problems in the mathematical sciences.Open Access funding provided by the Qatar National Library.Scopu
On the quantization of AB phase in nonlinear systems
Self-intersecting energy band structures in momentum space can be induced by
nonlinearity at the mean-field level, with the so-called nonlinear Dirac cones
as one intriguing consequence. Using the Qi-Wu-Zhang model plus power law
nonlinearity, we systematically study in this paper the Aharonov-Bohm (AB)
phase associated with an adiabatic process in the momentum space, with two
adiabatic paths circling around one nonlinear Dirac cone. Interestingly, for
and only for Kerr nonlinearity, the AB phase experiences a jump of at the
critical nonlinearity at which the Dirac cone appears or disappears, whereas
for all other powers of nonlinearity the AB phase always changes continuously
with the nonlinear strength. Our results may be useful for experimental
measurement of power-law nonlinearity and shall motivate further fundamental
interest in aspects of geometric phase and adiabatic following in nonlinear
systems.Comment: 4 figures, 11 pages, dedicated to Professor G. Casati on the Occasion
of His 80th Birthda
A nonrelativistic limit for AdS perturbations
The familiar nonrelativistic limit converts the
Klein-Gordon equation in Minkowski spacetime to the free Schroedinger equation,
and the Einstein-massive-scalar system without a cosmological constant to the
Schroedinger-Newton (SN) equation. In this paper, motivated by the problem of
stability of Anti-de Sitter (AdS) spacetime, we examine how this limit is
affected by the presence of a negative cosmological constant .
Assuming for consistency that the product tends to a negative
constant as , we show that the corresponding
nonrelativistic limit is given by the SN system with an external harmonic
potential which we call the Schrodinger-Newton-Hooke (SNH) system. We then
derive the resonant approximation which captures the dynamics of small
amplitude spherically symmetric solutions of the SNH system. This resonant
system turns out to be much simpler than its general-relativistic version,
which makes it amenable to analytic methods. Specifically, in four spatial
dimensions, we show that the resonant system possesses a three-dimensional
invariant subspace on which the dynamics is completely integrable and hence can
be solved analytically. The evolution of the two-lowest-mode initial data (an
extensively studied case for the original general-relativistic system), in
particular, is described by this family of solutions.Comment: v3: slightly expanded published versio
Dipole and quadrupole nonparaxial solitary waves
The cubic nonlinear Helmholtz equation with third and fourth order dispersion
and non-Kerr nonlinearity like the self steepening and the self frequency shift
is considered. This model describes nonparaxial ultrashort pulse propagation in
an optical medium in the presence of spatial dispersion originating from the
failure of slowly varying envelope approximation. We show that this system
admits periodic (elliptic) solitary waves with dipole structure within a period
and also transition from dipole to quadrupole structure within a period
depending on the value of the modulus parameter of Jacobi elliptic function.
The parametric conditions to be satisfied for the existence of these solutions
are given. The effect of the nonparaxial parameter on physical quantities like
amplitude, pulse-width and speed of the solitary waves are examined. It is
found that by adjusting the nonparaxial parameter, the speed of solitary waves
can be decelerated. The stability and robustness of the solitary waves are
discussed numerically.Comment: To appear in Chaos: An Interdisciplinary Journal of Nonlinear Scienc
Conservation laws, exact travelling waves and modulation instability for an extended nonlinear Schr\"odinger equation
We study various properties of solutions of an extended nonlinear
Schr\"{o}dinger (ENLS) equation, which arises in the context of geometric
evolution problems -- including vortex filament dynamics -- and governs
propagation of short pulses in optical fibers and nonlinear metamaterials. For
the periodic initial-boundary value problem, we derive conservation laws
satisfied by local in time, weak (distributional) solutions, and
establish global existence of such weak solutions. The derivation is obtained
by a regularization scheme under a balance condition on the coefficients of the
linear and nonlinear terms -- namely, the Hirota limit of the considered ENLS
model. Next, we investigate conditions for the existence of traveling wave
solutions, focusing on the case of bright and dark solitons. The balance
condition on the coefficients is found to be essential for the existence of
exact analytical soliton solutions; furthermore, we obtain conditions which
define parameter regimes for the existence of traveling solitons for various
linear dispersion strengths. Finally, we study the modulational instability of
plane waves of the ENLS equation, and identify important differences between
the ENLS case and the corresponding NLS counterpart. The analytical results are
corroborated by numerical simulations, which reveal notable differences between
the bright and the dark soliton propagation dynamics, and are in excellent
agreement with the analytical predictions of the modulation instability
analysis.Comment: 27 pages, 5 figures. To be published in Journal of Physics A:
Mathematical and Theoretica
GASE: a high performance solver for the Generalized Nonlinear Schrödinger equation based on heterogeneous computing
European Regional Policy in the Nord -Pas-de-Calais.
The Nord -Pas-de-Calais region benefits from rather considerable european funds. The eligibility of part of its territorry to objective 1 of the european regional policy brings new means to compensate for the delay in its development.Le Nord -Pas-de-Calais bénéficie de financements européens non négligeables. Notamment l'éligibilité d'une partie de son territoire à l'objectif 1 de la politique régionale européenne représente de nouveaux moyens au service du rattrapage de développement.Paris Didier. La politique régionale européenne dans le Nord -Pas-de-Calais. In: Hommes et Terres du Nord, 1995/3. La France du Nord dans l'Europe du Nord-Ouest : les nouvelles donnes et les infrastructures de transport. pp. 113-119
The initial-boundary value problem for the biharmonic Schr\"odinger equation on the half-line
We study the local and global wellposedness of the initial-boundary value
problem for the biharmonic Schr\"odinger equation on the half-line with
inhomogeneous Dirichlet-Neumann boundary data. First, we obtain a
representation formula for the solution of the linear nonhomogenenous problem
by using the Fokas method (also known as the \emph{unified transform method}).
We use this representation formula to prove space and time estimates on the
solutions of the linear model in fractional Sobolev spaces by using Fourier
analysis. Secondly, we consider the nonlinear model with a power type
nonlinearity and prove the local wellposedness by means of a classical
contraction argument. We obtain Strichartz estimates to treat the low
regularity case by using the oscillatory integral theory directly on the
representation formula provided by the Fokas method. Global wellposedness of
the defocusing model is established up to cubic nonlinearities by using the
multiplier technique and proving hidden trace regularities.Comment: 35 pages, 3 figure
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